Quantum Interference and the Spin Orbit Interaction in Mesoscopic Normal-Superconducting Junctions

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Quantum Interference and the Spin Orbit Interaction in Mesoscopic Normal-Superconducting Junctions

Keith Slevin, Jean-Louis Pichard, Pier Mello

To cite this version:

Keith Slevin, Jean-Louis Pichard, Pier Mello. Quantum Interference and the Spin Orbit Interaction

in Mesoscopic Normal-Superconducting Junctions. Journal de Physique I, EDP Sciences, 1996, 6 (4),

pp.529-555. �10.1051/jp1:1996228�. �jpa-00247201�

Quantum Interference and the Spin Orbit Interaction in

Mesoscopic Normal-Superconducting Junctions

Keith Slevin

(~),

^Jean-Louis Pichard (~>*) and Pier A.

Mello (~)

(~) ^Service de

Physique

de

l'ktat Condens4, CEA-Saclay,

91191 Gif-sur-Yvette, France

(~) Instituto de Fisica, UNAM,

Apartado

Postal 20-364, 01000 Mexico D-F-

(Received

18

July

1995, revised 14 November 1995, accepted 4

January 1996)

PACS.74.80.Fp

Points contacts; ^SN and SNS

junctions

PACS.74.50.+r

Proximity effects,

weak

links,

tunneling phenomena, and

Josephson

effects

PACS.72.10.Bg

General formulation of transport

theory

Abstract, We calculate the quantum ^correction to the classical conductance of _a disordered

mesoscopic normal-superconducting (NS) _junction

in which the electron

spatial

and spin

degrees

of freedom _are

coupled by

_an

appreciable

spin orbit interaction. ~fe _use random matrix

theory

to describe the scattering ⁱⁿ the normal part of the junction and consider both

quasi-ballistic

and diffusive

junctions.

The

dependence

of the

junction

conductance _on the

Schottky

barrier

transparency ^{at the NS interface is also} considered. We find that the quantum ^correction is

sensitive to the

breaking

of _spin rotation symmetry even when the junction is _{in a} magnetic

field and time reversal symmetry is broken. We demonstrate that this

sensitivity

_is due to

quantum interference between

scattering

_processes which involve electrons and holes

traversing

closed

loops

in the _same direction. We explain

why

such processes are sensitive to the _spin

orbit interaction but not to _a

magnetic

field.

Finally

_we consider the effect of the _spin orbit interaction _on the

phenomenon

of "reflectionless

tunnelhng."

1, Introduction

In mesoscopic

samples

of disordered normal metals at low

temperatures

it is

possible

to observe

a quantum ^correction to the classical Boltzmann conductance

ill.

Here

mesoscopic

means that within the

sample,

the interaction of _a

single

electron with other

degrees

of

freedom,

such _as other

electrons, phonons, magnetic impurities

etc., can be

neglected.

Such _a situation _can be realised in semiconductor and metal nanostructures cooled to milli-Kelvin

temperatures.

The

origin

of the

quantum

correction is

quantum

interference between time reversed

scattering paths. (By

"time reversed

path"

_{we mean} that the electron follows the _same

trajectory

but

m the

opposite sense.)

The correction is sensitive to the

breaking

of _spin rotation

symmetry

and is

suppressed by

the

breaking

^of time reversal

symmetry

_[2]. The

spin

orbit interaction has the

important property

^of

breaking spin

rotation

symmetry

^while

preserving

time reversal

symmetry.

As _an

example,

the two

probe

conductance G

=

(e~/h)g

of _a

quasi-one

dimensional _meso-

scopic wire can be

expressed

in the form _g

=

gel

+

bg

_[3]. The classical conductance of the _wire

is

O(N):

_gC~

=

N/(1+ s)

where N is the number of

scattering

channels and _s

=

Lli

is the

(*)

Author for

correspondence (e-mail: pichard©amoco.saclay.cea.fr)

©

Les

#ditions

_de

_Physique

₁₉₉₆

ratio of the

length

L of the wire to the elastic _mean free

path

I. In the absence of _an

applied magnetic

field the

quantum

correction

bg,

also called the "weak localisation

correction",

is

O(N°).

More

exactly,

in the diffusive

regime

1 _{< s} < N _we have

bg

=

-2/3

if the

spin

orbit interaction is

negligible

and

bg

=

+1/3

if it is not. In _a

magnetic

field sufficient _to break time reversal

symmetry

^the

quantum

^{correction is}

suppressed

and becomes

O(1IN).

More

recently,

_a quantum ^correction to the classical conductance of _a disordered _mesoscopic normal

superconducting (NS) junction

was observed [4]. ^{The correction} is most

pronounced

in

junctions

where the transparency ^{T of the}

Schottky

barrier at the NS interface is low T < 1 and the

length

of the normal part is such that Ts _m 1. Under these conditions _a

phenomenon

known _as "reflectionless

tunnelling"

_occurs. In [5] it is

suggested

that

quantum

interference is

responsible

for this effect. The

type

^of

scattering paths

involved in this

quantum

interference contain _a

segment

where the

path

of _an electron

(hole)

incident on the NS

boundary

is

subsequently

retraced

by

_an Andreev reflected hole

(electron.)

The enhancement of the

junction

conductance _can be _as much _as several times

greater

^than the classical conductance.

In this _{paper we}

investigate

the effect of the

coupling

of the spin and

spatial degrees

of

freedom of the electrons

by

an

appreciable spin

orbit interaction has _on this

quantum

correction.

As

previously mentioned,

this has _an

important

^effect on the

quantum

^{correction in} a normal metal. As the electron diffuses

through

the

metal,

the

spin

orbit

coupling

_{causes a} simultaneous diffusion of the direction of its

spin.

Interference between different

spin

states is now

possible

and this modifies the interference

responsible

for the weak localisation correction. What role

then,

if any, does this

spin

^diffusion

play

in the NS

junction?

At first

sight

the _answer to this

question

would be _{appear to} be _none at all. The

argument

is as follows: the

spin

rotations of _an electron and _an Andreev reflected

hole,

which traverse time reversed

paths,

_are

exactly _opposite

and cancel each other. Since interference between

scattering

_processes

involving

such

paths

_is believed to be

responsible

for the quantum cor-

rection,

it should be insensitive to the _spin orbit interaction. As _we shall demonstrate below this _is too

simple

and in fact the

spin

orbit interaction does affect the

quantum

correction to the classical conductance of _an NS

junction.

We have found the flaw in the

argument

^{is that}

it is not

only

_processes

involving

electrons and holes

traversing

time reversed

paths

which _are

responsible

for the quantum correction. Processes in which electrons and holes traverse

loops

m the _{same sense,} what _we shall call here identical

paths,

also contribute and in this _case the

spin

rotations do not cancel each other. The _most

striking

consequence of the existence of interference

involving

such

paths

is that the quantum ^correction is sensitive to the

spin

orbit interaction _even in _a

magnetic

^field.

In the normal metal _we have _seen that the

quantum

^correction to the classical conductance

is

suppressed

when time reversal symmetry is broken in _an

applied magnetic

^field. This _is

easily

understood since the symmetry between time reversed

paths

is broken when the time reversal

symmetry

is broken. In the NS

junction however,

the contribution of identical

paths

is still present even m a

magnetic field;

the electron and hole carry

opposite charges,

and

so the Aharonov-Bohm

phases

that

they

accumulate _as

they

follo1&~ such

trajectories

cancel.

Moreover,

_since the electron and hole

undergo

identical spin

rotations,

as

they

follow identical

paths,

this contribution is sensitive _to the

spin

diffusion induced

by

the

spin

orbit interaction.

The paper is

organised

as follows. In Sections 2-4 _1&~e deal with _some necessary

preliminaries:

the

Bogolubov

de Gennes

equations, scattering theory

and conductance formulae for the NS

junction, generalising

the standard

theory

to take into account the _spin orbit interaction.

In Section 5 _1&~e

investigate

a

quasi-ballistic

NS

junction,

that is to say a

junction

whose

length

is shorter than the _mean free

path

but

long enough

_so that its

scattering

matrix _is well described

by

_a certain random

phase approximation

_[6]. We express the

junction

conductance

Table I. The various ensembles

for

which

GNS

is calculated. The abbreviation TRS _means time reversal

symmetry

^and SRS

spin

rotation

symmetry.

Ensemble TRS SRS

Orthogonal

_{yes yes}

Unitary

I broken _yes

Unitary

II broken broken

Symplectic

_yes broken

Table II. The

quantum

correction

bGNs

"

(2e~/h)bgNs for

the

quasi-ballistic

NS

junction

for

the ensembles listed in Table I.

Ensemble

bgNs

T

= 1 T _< I

Orthogonal/Symplectic -2~Tsf(r)

-4Ns

+Nsr~

Unitary

-2s

f(r)

-4s

+sr~

Unitary

II _+s

f(r)

+2s

-sr~ /2

GNS

_{as a sum} of _{t~vo terms}

GNS

_"

~)

^{gNs "}

~) (gis

₊

_~~NS)

_(~~

a classical conductance

g(~

and _a

quantum

correction

bgNs

due to

quantum

interference. We determine the

quantum

^{correction for the four ensembles listed in Table I and the results} are

presented

in Table II. In Section 6 _we present a semiclassical

interpretation

of these i«esults which

permits

us to

identify

the

type

^of

scattering paths

which interfere _to

produce

the

quantum

correction. In

agreement

with reference

[5],

we find that there _{is an}

important

contribution due _to interference between processes

involving

electrons and holes

traversing

time reversed

paths,

but that there is also _a second contribution from processes

involving

electrons and holes

traversing

identical

paths.

We

explain why

this contribution is sensitive to the

breaking

of spin rotation

symmetry

but not to the

breaking

of time reversal

symmetry.

After

reaching

_a firm

understanding

^of the

quasi-ballistic junction

_we consider the diffusive

regime, looking

first at

junctions

without _a

Schottky

barrier _at the NS interface _m Section 7.

In _zero

magnetic

field the quantum ^correction is known to be of

O(N°) _[7-9].

In _a

magnetic

field the authors of reference [10] find that the correction

though

smaller _is still

O(N°).

The effect of the spin orbit interaction _was not considered in reference

[10j

^and so we repeat ^their calculation

taking

it into account. We find that

breaking

_spin rotation

symmetry multiplies

the correction

by

_a ^factor of minus _one

half, regardless

^of whether time reversal

symmetry

is

broken _or not.

In Section 8 _we discuss the dramatic enhancement of the

junction conductance,

kno1&~n _as

"reflectionless

tunnelling,"

which is observable when r _< 1 and rs _m 1. Under these conditions the contribution of time reversed

paths

_is

O(N)

and dominates that of identical

paths

which

is

O(N°)

_so that the reflectionless

tunnelling

effect

is,

m a first

approximation,

insensitive to the _spin orbit interaction. This conclusion has been confirmed

by

_carrying out _a numerical

simulation of _a

junction

under the relevant conditions.

2~

Bogolubov

de Gennes

Equations

and the

Spin

Orbit Interaction

The

Bogolubov

de Gennes

(BdG) equations

_[11]

appropriate

^for a metal where the electrons'

spatial

and

spin degrees

of freedom _are

coupled by

_a

significant spin-orbit interaction,

are

l~~ ^~*'

_j-~^~ _j-

_~h

^{~/,e " f}

_~h

^~fie

⁽²⁾

' ~

where

He

=

Ho EF

₊ Q

Here

Ho

e

Ho (r,

_a,

p)

is the

single

electron Hamiltonian of the metal

incorporating

the _spin- orbit interaction and

EF

_is the Fermi energy. In

deriving

this

equation,

it is assumed that _an

attractive point ^like interaction exists between the electrons:

The full

interacting

Hamiltonian is reduced _{to an} effective

non-interacting

Hamiltonian

by introducing

effective

potentials

~(r)

=

%*(r, I)*(r,1) (4)

Q(r,

_a,/~)

=

lfi(1 2ba,~)*lr, a)*t(r,

_{/~) 15)}

where the overline indicates _a thermal average with respect to the Fermi distribution function and §/ is the usual field

operator

_appearing ^{in the} second

quantised

formulation of the inter-

acting

^electron

problem.

The time reversal

operator T,

which appears in the BdG

equations

has the form

T =

pC (6)

p =

Say

=

1° _(~ ii)

~~

with C the

operation

of

complex conjugation.

The

eigenstates

of the BdG

equation

describe the excitations of the

interacting

electron

system.

The

meaning

of the electron _ifi~ and ifi~

wavefunctions _can be _seen

by writing

the field operator as

*lr, a)

=

~j _{(ifi[lr, a)n~~} Tlifillr:

a)In~)j

¹⁸⁾

where _{~n is} the annihilation operator ^{ofthe excitation labelled} n. The

corresponding eigenvalue

en of the BdG

equations corresponds

to the energy of the excitation. With the aid of

(8)

the

occupation

^of the

eigenstates

^of

He

for _a

given

excitation _can be determined.

3.

Scattering Theory

for the NS Junction

In this section _1&~e

develop

the

scattering theory appropriate

to the normal

superconducting junction

shown

schematically

in

Figure

1. We treat the

scattering

at the NS interface within

Andreev's

approximation.

We _suppose that _any

impurities

in the

system

_{are m} the _region indicated

by shading

in

Figure 1,

^and we make the

approximation

that the

magnetic

field is

zero

everywhere except

ⁱⁿ this disordered region. This _is reasonable for the low fields of interest which affect the interference between electrons and holes _m the normal part ^{of the}

junction.

NORMAL SUPERCONDUCTOR MBTAL

RANDOMPOTBNTIAL SCHOTTKYBARRW

Fig.

1. A schematic of the NS

junction

for which the scattering

theory

is

developed

in Section 3.

3.I. SCATTERING MATRICES _FOR ELECTRONS AND HOLES. To facilitate the

explanation

of the

formalism,

it is

helpful

to

develop

the

scattering theory

for _a definite model. We have chosen _a lattice

tight binding model,

the _same model which _we will _use later _m numerical

simulations. An exact

analogous explanation

is also

possible

for _a continuum model.

We consider _a cubic lattice and take into account nearest

neighbour

interactions

only.

We denote

by ~~(x,

_{y, z,}

a)

the

amplitude

that the electron is in _{an s-} orbital _at

lx,

_y,

z)

with

spin

a and

similarly

ifi~

lx,

_{y, z,}

a)

for _a hole. We include in the Hamiltonian _a

spin

orbit term which arises from the Zeeman

coupling

of the electron

spin

with the effective

magnetic

field felt

by

the electron _as it _moves in the

spatially varying potential

of the lattice. We

ignore

the direct Zeeman

coupling

of the

spin

to any external

magnetic

field. A

simple

calculation shows that the Hamiltonian has the form:

< X>Y> Z,

a(tie(I,

_Y,Z,JL > _"

E0~a,v

< x, Y, z,

alHe lx i,

_{Y, z,} J1> =

vS,~

< x, v, z,

alHe lx, i,

_{z, J1 > =}

vl,~

~~~

< X> l/, Z,

ajHelX,

_l/>Z

I,

_/l > _#

eXp(-S£XX) Uj,~

where

vj

~ ⁼

l§i~,~ _vii[a~]~~~

U$)v =

l§b«,v Vii(ay]a,v (10)

vj,~

=

l§i~,~ vii[az]~,~

An external

magnetic

field

B, applied

in the +y direction is modelled

by

Peierl's factors in the matrix elements between nearest

neighbours

m the _z direction. If _a is lattice constant a =

27rBa~ /~o

where

~o

=

hle

_is the flux

quantum.

^The transverse dimensions are1 < _x <

L~

and 1 < y ^<

Lg.

The Hamiltonian

(9)

_can be

regarded

_{as a} three dimensional

generalisation

of that

proposed

in reference

[12]

as a model for _a two dimensional electron gas formed at the surface of _a III-V semiconductor.

The relevant energy scale of the model _is determined

by (~

₊

_l§~.

For convenience _we shall set this _to

unity

^with the choice

l§

_{= cos} 6

Vi

_{= sin} 6

(11)

Varying

the

angle 6,

_{we may} set an

arbitrary

ratio of normal

potential coupling

_Vo to spin orbit

coupling Vi,

while

keeping

the extent in energy of the

density

of states

roughly

constant. ivith this

choice, u~,uY

and u~ _are all elements of

SU(2). Using

the

homomorphism

of

SU(2)

with the three dimensional rotation group

SO(3),

_{we can}

interpret

the u's _as rotations of the _spin of the electron _as it _moves between _nearest

neighbours _[13].

A

product

of nearest

neighbour

matrix elements

along

_a

path

will have the form

exp(14l)u

where 4l is the Aharonov Bohm

phase picked

_up

by

the

electron,

and _u E

SU(2)

is the rotation of the electron's

spin

_as it traverses the

path.

The Hamiltoman

He

has the form

given

in

(9) everywhere except

in the disordered

region

located in 0 _{< z <} L.

There,

_{some or} all of the Hamiltonian matrix elements _are

supposed

random. In reference

[12]

^the

diagonal

elements _were assumed to be

independently

and identi-

cally distributed,

while the

parameter

⁶

controlling

the _spin orbit interaction _was held fixed. In

references

[14,15]

a random

spin

orbit interaction _was also considered. For the

present

purpose

we do not need to

specify

the

precise

distribution.

First,

_we consider the

scattering

^{of electrons} incident at _an energy E

=

EF

+ e. To the left

of the disordered section _we

expand

the electron wavefunction _{ifi~ in} terms of the Bloch states of

(9)

with energy E.

~~(x,

_{y, z,}

a)

=

~j a+n~+»(x,y, a) exp(+ik~z)

₊

~j a-~ifi-»(x,y, a) exp(-ik~z)

n;Zmkm<0 tZmkm=0

(12) ifi-~(x,Y,a)

=

£P~,~ifi+~lL~

x +

i,Y,J1) l13)

~

As _z

~ -cc, far from the disordered

region,

_we

impose

the

boundary

condition that the allowed

states consist

exclusively

of

incoming

and

outgoing propagating

_waves. Thus in _{z <} 0 states

with Zm

k~

> 0 _are excluded. We denote

by

2N the number of

"open channels",

I-e- states with Zm

kn

= 0 that carry a

positive probability

current in the +z

direction;

there _{are an}

equal

number

carrying

current in the _-z direction. We label these states _so that +n _{carries a} current

m the _+z direction and _{-~ a} current in the _-z direction. After _a suitable normalisation of

the _transverse wavefunctions

(see Appendix A)

the electric current due to electrons at the left

of the disordered section is

Ie

"

) _{~j l~+~l~ l~-~l~ (14)}

~,Zmkm=0

The

boundary

condition _{as z ~ -cc,}

imposed above,

ensures that

only

_open

channels,

and not "closed channels" with Zm

k~ # 0,

contribute to the current. A similar

expansion

_may be made _on the

right

in terms of _a set of coefficients

(a[~, a[~)

with the

boundary

condition that

~ve admit

only

those states ~vith Zm

k~

> 0.

Thus,

far to the

right

^of the disordered

section,

as z ~ +cc, the allowed states again consist

exclusively

of

incoming

and

outgoing propagating

waves.

The 4N _x 41V

scattering

matrix for electrons

Se

relates the 4N

incoming

flux

amplitudes

at the

left,

_a+

=

(a+»,Zm

_km

=

0)

and the

right a[

=

(a[~,Zm

km =

0)

with the 4N

outgoing

flux

amplitudes

at the left _a-

=

(a-»;Zm kn

=

0)

and the

right a[

=

(a[~,Zm

_km

=

0)

Se ~/

"

~T (15)

~- ~+

The matrix

Se

has the structure

s^~

l~e

_te ^~~

^jilt)

T~

m terms of the 2N _x 2N reflection and transmission matrices for left incidence

(r~,t~)

and

right

incidence

(r[, t[).

Since _{we are}

considering

time

independent scattering,

the currents to the left and

right

of the disordered section _must be

equal,

^{and therefore} it follows that S~ is unitary. ^There is _an

additional restriction _on

Se when,

in the absence of _an

applied magnetic field,

the Hamiltonian is time reversal invariant ie.

[He,

_{2~j =} 0 with T given m

equation (6).

For _a suitable choice of

transverse wavefunctions

(see Appendix A) Se

will then

satisfy

~~ ~~ _-~2N ~~ ~~ ^~N

~~~~

where

12N

_means the 2N _x 2N _unit matrix. This _can be written in the

equivalent

form

re =

-r) r[

=

-(r[)~ ₍₁₈₎

te _"

+(t[)~

The

simplicity

of these

relations, compared

with for

example

^{those of reference}

[16],

is related to the presence of _p in

equation (13) (see Appendices

A and

B)

We _now turn to the

scattering

matrix

Sh

for the holes. From the BdG

equations

_{we can see}

that the hole wavefunction ifi~ evolves

according

to

Hh~~

" ih

~

ifi~

(19)

where

Hh

is given

by

Hh(+B]

=

TH~[+B]T

=

-He j-B] (20)

If _~fi~ describes the

scattering

of _a hole with excitation energy ^+e in _a field +B then

T~fi~

describes the

scattering

of _an electron at energy -e also in field +B. We shall make _use of this in two ways.

Firstly,

outside the disordered

region

B

= 0 and

Hh

=

-He.

Thus outside the disordered

region

it is useful to

expand

_~fi~ in terms of the Bloch states of

He

at energy

EF

_e. At the

left,

for

example

ifi~(x,

_{y, z,}

a)

=

£ b-»ifi+»(x,

_y,

a) exp(+ik»z)

₊

~j b+»ifi-»(x,

_y,

a) exp(-ik~z)

~;Zm km <0 «Zmkn=0

(21)

Note that _since in this

region Hh

"

-He

the

probability

currents are reversed

by comparison

with the electron _case. We therefore associate the coefficient

b+~,

the flux

amplitude

for _a

positive

hole

probability

current in the _+z

direction,

with the wavefunction

proportional

to

exp(-ik~z).

The holes _{carry an}

opposite

^electric

charge

to that of the electrons _so that

they

carry an electric current

Ih

=

~

~j _{(b+~[~ [b-~[~ (22)}

~

wZmkn=0

By definition,

the 4N _x 4N matrix

Sh

relates incoming hole

probability

currents to

outgoing probability

currents

Sh I)/

=

)/ ₍₂₃₎

+

The matrix

Sh

has _{a structure} similar _to that of

equation (16);

_1e.

~h

"

~

~) (2~)

h ~

Secondly by rewriting Tifi~

in _terms of electron flux

amplitudes,

and

recalling

that these

amplitudes

_are related

by S~,

we arrive at _a relation between

Se (-e, +B)

and

Sh(+e, +B)

sh(+f,+B)

"

~f _~~~ sll~f,+B) ~j~

1)~ ¹²⁵⁾

or

rh(+e,+B)

=

-[r~(-e,+B)]*

~ll+f>+B)

~

~l~ll~f,+B)l~

th(+e,+B) j~~~

=

+[te(-e,+B)]"

t[(+e,+B)

=

+(t[(-e,+B)]*

Again

_{we assume} here _a suitable choice of transverse wavefunctions

(see Appendix A) Following

a similar line of

argument

^{it is}

possible

to demonstrate _a

relationship

between

Se (+e, +B)

and

Se (+e, -B).

3.2. ANDREEV SCATTERING AT THE NS INTERFACE. In this section _we outline the cal-

culation of the coefficients of Andreev reflection

(17]

^{at the NS interface. These} are

essentially unchanged by

the introduction of _a

spin

orbit interaction in the materials

forming

the

junction.

We _assume

throughout

that

/lo

<

EF,

_a condition which is realised in

practice.

Far from the NS interface _m the normal metal the

superconducting

_gap /l _~ 0. On the other

hand,

^far from the NS interface in the

superconductor,

/l _~

/lo exp(i~)

where

/lo

is real. In

general

the reflection coefficients will

depend

_on the

precise

form of /l in the transition

region

near the

junction.

For _a

point

contact

junction, however,

it is

permissible

to _{assume a}

simple step

^model

=

~o _exp(i~)

^~~~~

We _are interested in the situation where the _energy E

=

EF

_{+ e} of the incident electron _is in the _{energy gap} of the

superconductor EF

_< E _<

EF

₊

/lo. Anticipating

somewhat in order to avoid unnecessary

algebra,

we find the electron is

mainly

reflected _{as a} hole like excitation. A

solution of the BdG

equation

in the normal metal

corresponding

to this is

l~~

^"

~XP(S~~~Z)~fi~(X>Y>~)

~

~( ^~XP(S~~ ~Z)~fi~~(X>Y,~) (2~)

e

where

r)~

^{denotes the} matrix of Andreev reflection

amplitudes

and the

superscript (+)

refer to Bloch states with energies

El+)

=

EF

+ _e. The first term describes _an excitation where

an electron above the Fermi level is incident from the left in channel _n. The second term

corresponds

to _an excitation in which _an electron below the Fermi level is

annihilated,

_i-e-

to _a reflected "hole" with

opposite velocity

and _spin to that of the incoming electron. Since

/lo

<

EF

_{we can} to _a

good approximation

_ignore the difference between

k(~~

and

k(

_and

similarly

the differences between the transverse wavefunctions.

Requiring

that the wavefunction and its derivative be continuous at the

boundary

of the

superconductor

leads to

r)~

=

iexp(-i~)

,

r$~

₌

iexp(+i~) (29)

Here _we have made the further

assumption

that _e <

/lo,

the limit of interest in what

follows,

and _we have also _given the reflection coefficient for _an incident hole.

3.3. THE SCOTTKY BARRIER AT THE MS INTERFACE. in real MS

junctions,

_a mismatch between the conduction bands of the two materials which make _up the

junction

^results in the creation of _a

Schottky

barrier at the interface. This barrier

plays

_an

important

role in the

physics

^{of the} device and _{so we} must take it into account. We shall model the

Schottky

barrier _as

a

simple planar potential

barrier. At the barrier _an incident

particle

_may be either transmitted without _a

change

of momentum _or

specularly

reflected. We

neglect

_any

dependence

of the reflection and transmission

probabilities

_on the momentum of the incident

particle

_so that the

properties

of the barrier _are described

by

_a

single

parameter ^r E

(0,1],

its

transparency.

^The transmission and reflection matrices which make _up the electron

scattering

^matrix

St

of the barrier _are

tf

=

/f _12N

#

~r _~~

_~~~~

r[~

=

-fit _Ml

The

precise

form of the 2N x 2N matrix _MB

depends

_on the choice of _transverse wavefunctions.

For

subsequent analysis

_we need

only note, however,

that _MB is

antisymmetric

and

unitary.

The hole

scattering

matrix

St

for the barrier is related _to

St

_in the usual _way

by (26).

3.4. COMBINATION _oF SCATTERING MATRICES. It is the purpose of this

section, having

considered above the

scattering

matrices for the various

components

^of the NS

junction,

to

explain

how the

scattering

^matrices _may be combined to find the total

scattering

^{matrix. We} consider first _a

junction

without _a

Schottky

barrier. For the normal part we can write

"~~~~~

~ =

~e ~

etc. ~~~~

0 _rh

are 4!V-dimensional matrices and

C+ ~

, C+ ~ ~i

(33)

~~

^'

~

4N-dimensional vectors, ⁱⁿ the notation of Section 4. For the Andreev

part

we define the 4N _x 4N

scattering

matrix

S~ by

S~c[

=

c[ (34)

By

reference to Section 3 this has the form

~ ~

r)~12N

0 ~~~~

For the combined

system,

^{the 4N} x 4N S matrix _is defined

by

Sc+

₌ c-

(36)

In

equation (31)

_we

replace

CL in terms of

c[

as in

(34);

_we then

perform

the matrix multi-

plication and,

from the two

resulting equations,

eliminate

c[.

From the

expression relating

_c+

and _{c- we} extract the 4N _x 4N S matrix of

(36)

to obtain

S = r +

t'S~

~,~~

^t

⁽³⁷⁾

In the electron-hole _spaces, S of

equation (37)

has the _structure

S=

~~~

_~~~

(38)

The rhh

re~ etc.

being

^2N x 2N matrices. To determine the conductance _1&~e shall need the submatrix The From

equations (37)

and

(38)

_we find

Using

the structure

(35)

of

S~

we can write _The as

~~~ ~ ~~~~~

l r'SA

~~

~~ ~~

For any

nonsmgular operator

^D we now use the operator

identity

_(18]

1

Ii

~

(41)

D

ee

l~ee ~e~ ~~~~

ee

~~ ~~~

The "

ti~te _l~ ~[~te~i~shl

^~_~~

l~~~

For _a

junction

with _a

Schottky

barrier _we first consider the

composition

of the

Schottky

barrier and the

superconductor.

The

scattering

matrix for this system can be obtained from

equation (37)

where _r, t and t' _are taken from the model

(30)

for the barrier. With the aid of the

identity (41)

and the

following

_one

(18]

l~

he

~h ~~~

Dee

~h

~Dhe

~~

~~~~

~~ ~~~

~BS

^{T~S ~BS}

"

()

^eh

~he

T~)

~~~

l

~XP(~S~) (~/(2 ~)j

_12N

~fS

₌

_j~[BS]

~~

~~i

" 2

~/(2 ~)j

_WB ^~~~~

~BS

_j~[BS]

_~~

hh ee

The

scattering

^{matrix for the}

complete system

^of normal part,

Schottky

barrier and _supercon- ductor _{can now} be obtained from

equation (37),

where

r',

etc. refer _to the normal metal _as

before,

but

S~

is

replaced by S~~

of

(45). (Note

that in

deriving Eq. ₍₃₇₁

the structure

(35)

of

S~

1&~as not

used.)

4. Conductance Formulae

We _assume that the bias

voltage

is small in comparison ^to

/lole,

and that the _size of the super-

conducting

part ^is

long enough

_so that there is _no

quasi-particle

current in the

superconductor.

In this _case, the _zero temperature ^dc conductance

GNS

of the

normal-superconducting junction

can be described

by

_a

simplified

"Landauer" formula which has been derived in reference

(19-21]

where _The is the 2N _x 2N matrix of electron-hole reflection

amplitudes

for the

composite

system.

There is _an

important simplification

if the Hamiltonian of the

system

is time reversal invari- ant and the bias

voltage

is small in

comparison

to the Thouless _energy

Ec [22],

so that the

energy

dependence

of

Se

_can be

neglected.

The conductance then has the form

[23]

where

Tn

_are the

eigenvalues

of

tet).

_{We have verified that this result still holds when}

_spin

rotation

symmetry

is broken

by

the

spin

orbit interaction.

In what follows _we wish to compare the

quantum

conductance of the NS

junction,

calculated from

(46),

with the classical conductance of the

junction.

This latter

quantity

is determined

using the classical rule of

combining

conductances

i/g

=

i/gi

+

1/g2 148)

This

corresponds

to the addition of flux intensities _as

opposed

to flux

amplitudes.

The conduc-

tance associated with the electron

traversing

the normal

part

is

2N/s

and

similarly

for the hole

m the

traversing

the normal part ⁱⁿ the

opposite

direction. The conductance of the barrier is

2N[r))[~,

_so that the classical conductance _is

~j~~j~BSj2

~~~

1 ₊

2s~~))

_[2 ^~~~~

The classical conductance _is insensitive to the

breaking

^of time reversal and _spin rotation

symmetries.

5.

Quasi-Ballistic

Junction

In this section _we shall calculate the conductance of _an NS

junction

to first order _{m s}

=

Lli,

where L is the

length

^of the normal part ^{of the}

junction

and I is the elastic _mean free

path.

We

neglect

terms of order

s~

_{and above}

so that result is

strictly applicable only

_m the limit that _{s <} 1. Nevertheless _we shall _see that the results of the calculation shed considerable

light

on the

origin

of the

quantum

interference in the device.

The 2N _x 2N reflection matrix _The for the

system consisting

^of the normal

metal,

barrier and

superconductor

_can ^{be obtained} _as discussed in Section 3.4. After

expanding

to second order _m

r[, r[,

which is sufficient for _an evaluation of _gNs to first order in _{s, we} have

~he ~~~~~~~ ^~

~~~~i~~~ii~e

+

~~~~/~~~~~~e

+

~~~~~~~~ii~~Tii~e

(50)

+

t[r))r[r))r[rf~~t~

₊

t[r))r[r))r[r))t~

₊

t[r))r[rf)r[r))te

₊

The conductance _is found

by substituting

this into

(46)

and

performing

_{an average over an}

ensemble

ofscattering

matrices

Se, describing

_a set

ofmicroscopically

^different but _macroscop-

ically equivalent configurations

of

impurities

_m the normal part of the

junction.

In

principle,

the distribution for

Se

should be calculated from _some model distribution of Hamiltomans. We shall

not, however, attempt

to do that here. Instead _we will _assume that the

resulting

ensemble of

scattering

matrices

Se

is distributed

according

to the "local maximum

entropy

^model"

[3, 24, 25].

If the

geometry

is

quasi-1d

and the number of channels N

sufficiently large,

then results obtained with the aid of the local maximum

entropy

^model are known to be identical to those obtained from the class of

microscopic

models described

by

the nonlinear

sigma

model

[26, 27].

The distribution of

Se

_m the local model

depends

_on

N,

s and the

symmetry

of the Hamil-

tonian,

I.e. whether _or not time reversal

symmetry

^{is broken and whether} or not

spin

rotation

symmetry

is broken. There _are

four

ensembles

(Tab. I.)

The critical

strengths

of the

magnetic

field and the spin orbit interaction

separating

the various ensembles should be similar to those associated with the weak localisation effect in normal metals. The details of the distribution of S~ for the four ensembles _can be found in

Appendix

C.

It will be

helpful,

when _{we come} to discuss the

interpretation

of the

results,

to write

(50)

in the form

~

The "

~

_Pi

₍₅₁₎

1=1

Each term m the series

represents

the contribution of _a

particular scattering

_processes to The- The classical conductance is obtained

by ignoring

interference between different _processes and

summing

intensities

cc

g#s

= tr

i=1 ~j _{p~p) (52)}

The quantum correction to this classical conductance is found

by

_summing the interference between different _processes

bgNs

" tr

i#j ~j _{p~pj (53)}

After _carrying out the average we find that

g~8 2~'T~~'~(1~ _2'T~~'~S

_~

°(S~)) (5~)

which agrees with the

expansion

of

(49)

to the order _{we are}

considering.

^The

explicit expressions

for

bgNs

_are collected

together

_m Table II. The function

f

of the barrier

transparency,

which

appears in the

table,

has the

explicit

form:

fir)

-

~~~l~~ i~~~l~~

ii ¹⁵⁵⁾

There _are two obvious

limiting

_cases: T ₌ 1

corresponding

to _a

junction

without _a

Schottky

barrier and r <

corresponding

to _a

junction

with _a

high Schottky

barrier. The first

point

to note is that the

quantum

^{correction is of}

O(N),

the _same order _as the classical

conductance,

m zero field.

Secondly

for T <

1,

the conductance increases _as disorder _is added to the

junction.

This is the _essence of the dramatic reflectionless

tunnelling

effect which _we discuss

m Section 8.

Thirdly

when time reversal

symmetry

is broken

by

the

application

of _a

magnetic

field the

quantum

correction is

O(N°

_{and not}

O(1IN)

_as

might

have been

expected by analogy

with the weak localisation effect in _a normal metal. The

final,

and

perhaps

the _most

surprising

result,

_is that in _a

magnetic

^field the

breaking

of

spin

rotation

symmetry by

the _spin orbit interaction

multiplies

the

quantum

correction

by

_a factor of _{minus one} half _even

though

the

symmetry

^{of the}

Hamiltonian,

_m the _sense of random matrix

theory,

is

unchanged

and remains unitary.

xmx

NORMALMBTAL SUPERCONDUCTOR

Fig.

2. An

example

of _a

path

which contributes to process pi

corresponding

to _an electron

(solid linel

_traversing the normal part ^{of the}

junction

whose

path

is then retraced

by

the Andreev reflected

hole

(dashed line).

6. Semiclassical

Interpretation

The

importance

of

quantum

interference between _processes in which the

path

of _an electron

(hole)

incident _on the NS

boundary

is

subsequently

retraced

by

_an Andreev reflected hole

(electron)

_was first

pointed

out in [5]. ^In the absence of _a

magnetic field,

and if the bias

voltage

is small

enough,

electrons which _move

along

_a

path

in _one

given

_{sense are}

phase conjugated

with holes

traversing

the time reversed

path.

In _a

magnetic field,

_or if the bias

voltage

is

large enough,

this

phase conjugation

_is

destroyed.

However _we have _seen that there is _a

significant quantum

^correction even in _a

magnetic

field. There must therefore be _an additional _source of

quantum

interference which is not sensitive to the

breaking

of time reversal

symmetry.

^As

we shall _see the relevant _processes involve

paths

m which _an electron

(hole)

and _an Andreev reflected hole

(electron)

traverse a

loop

in the _{same sense.} In order to remain concise, ^we shall refer to such processes as

containing

identical

paths.

The interference

involving

such

paths

_can

only

be

destroyed by applying

_a

large enough

bias

voltage.

The

physical importance

^{of the}

bias

voltage

_{as a}

"symmetry breaking parameter"

is discussed in

[10].

Only

_some of the terms in

(53)

_are found _to be _nonzero after _{averaging, so} that in fact

bgNs

₌ tr

(pip)

⁺

^Pip(

⁺

^psp)

⁺

_p7p()

⁺

^{Ols~) 156)}

where pj means the

jth

term in

(50.)

Consider first the interference between _{process pi} and p5. An

example

of _a

scattering path

which contributes to process pi

Pi "

t[~))te ₁₅₇₎

is illustrated in

Figure

2. Since _{we are}

working

_to first order in _s it is sufficient to consider the motion of the electron and

holes, traversing

from _one side of the normal

part

of the

junction

to the

other,

_as

ballistic,

_so these

trajectories

_{appear as}

straight

lines _m

Figure

^2.

^Examples

of

paths

which contribute to p5

P5 "

t~Tf~T~Tf)~~~iite (~~)

are illustrated in

Figures

3 and 4. To first order _{m s} the interference between processes pi

X=Impufity

x«xb

x=xa

NORMALMBTAL SUPERCONDUCTOR

Fig.

3. An

example

of _a "time reversed

path"

which contributes to process p5. The

path

of the

electron moving from _xa to _xb is retraced by the Andreev reflected hole _as it _moves from _xb to _xa.

Interference between this

path

and that illustrated in

Figure

2 is insensitive to the spin orbit interaction and is

suppressed

in _a

magnetic

field.

X=ImpuTity

X"Xa"Xb

NORMALMBTAL SUPERCONDUCTOR

Fig.

4. An

example

of _an "identical

path"

which contributes to the process p5. The electron _moves around _a

loop

and returns to _~a. The Andreev reflected hole traverses the

loop

_m the _same direction.

Interference between this

path

and that illustrated in

Figure

2 is sensitive to the

spin

orbit interaction but not to a

magnetic

field.

and _{p5 is}

tr

(pip)

⁺

_p5P))

"

2[r))[~[rf~~[~tr(r[r[) (59)

The

remaining

terms

involving

_pi and p7 contribute

tr

(piP~

⁺

p7P))

^"

-2[r))[~tr (r[r() (60)

In the interest of

brevity

_we will concentrate on the interference between pi and p5. A very similar

analysis

_is

possible

for the interference between _{processes pi} and _p7

leading

to identical

conclusions. We

proceed by relating

the

product

^of electron and hole reflection matrices _m

(59)

to a

product

of _an electron and _a hole Greens function. To

simplify

the

algebra

_we shall

suppose that both the _spin orbit interaction and the

magnetic

field _{are zero}

everywhere

except

m the disordered

region.

We shall also _suppose that

Ly

₌ 1 and

impose periodic boundary

conditions in the _x direction. The Bloch states at energy E have the form

1fi2m(x,z,a)

=

exp(ikj~x)exp(ik2mz)b~~~~~~

a2m "

I

~fi2m+1(X,

Z,

a)

"

eXp(Sk(m+lX) eXp(Sk2m+1Z)b~,~2»1+1

~2m+1 "

I

(61) ifi-2m(x,z,a)

₌

-exp(-ik(~x)exp(-ik2mz)b~,~_~~

_{a-2m =}

~fi-(2m+1)(X,Z,a)

"

eXp(-Sk(m+lX)~XP(~S~2m+lZ)~a,a-(2m+1)

~-(2m+1)

i

~~~~~

k(~

=

k(m+1

"

)~°

"

~''' '~~

_~~~~

and the energy and the momenta are related

by

E ₌ 2 _cos

kc

+ 2 cos

km (63

The reflection matrices for electrons and holes _can be related to the

corresponding

^Green's

functions _as indicated in

Appendix

A.

lr~)m~

₌

-i~exp(+i(km+k~)L)

~j ~fi~m(Xba)G$ (Xb,

L> ~l Xa L>

~')~fi-~ (Xa~') (64)

xaiba'

[r[]~m

=

-i~exp(-I(km+k»)L)

~j ifi[~(xba)G)(xb,L,a;xa,L,a')ifi+m(xaa') (65)

~~~~~aJ

Note

that,

for

convenience,

the states

(61)

have _not been normalised to carry identical currents, proper ^account of this has been taken in the expressions

(64)

and

(65)

for the reflection matrices.

The

quantum

correction involves _a trace _over the

product

of _r~ and _rh.

Using

the relation with the Green's function this _can be

separated

into two elements: _an

integration

_over the _cross section

involving

the transverse wavefunctions and _an average of the

product

of _an electron and _a hole Greens function. We will consider the second element first. This involves

evaluating

A~_~,~

~m

(Xa, Xb)

"

(~i~(Xa> L,

_{a-n, Xb,}

L, a+m)G$(Xb, L,

_{a+m, Xa,}

L, a-n)) (66)

Within the semiclassical

approximation,

_{as is}

explained

in reference

[28], Chapter

12 and

13,

we can express the Greens functions _as summations _over

paths:

G~(Xb, L,

_{a+mi Xa,}

L, a-n)

"

£~=~a-~b ^~J ~XP(S~J

^~

S~J)iUJj?+»i

?-n

(fi?)

G~

_~~b>

L,

_{a-n, Xa,}

L, a+m)

"

£j:~a-xb ~J ~XP(~S~J S~J (UJj?-n,?+»1

Substituting (67)

into

(66)

and

taking

the disorder _{average we} find that

only

two contributions

remain:

6.I. TIME REVERSED PATHS. The

path

of the electron

moving

between _xa and _xb is

retraced

by

the Andreev reflected hole _as illustrated in

Figure

3. The electron and hole

charges

are

opposite

but

they

traverse the

path

in

opposite

directions so the Aharonov-Bohm

phase

factors

they

^accumulate do not cancel each other _out. Thus this _{term is}

important only

when the

magnetic

^field is

negligible

and the Hamiltonian _is time reversal

symmetric.